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INVARIANT TWINS
subject to revision and rejection; the theory of invariance as a
permanent addition to pure mathematical thought appears to
rest on firmer ground.
Another of the ideas originated by Cayley, that of the geo-
metry of 'higher space* (space of n dimensions) is likewise of
present scientific significance but of incomparably greater
importance as pure mathematics. Similarly for the theory of
matrices, again an invention of Cayley's. In non-Euclidean
geometry Cayley prepared the way for Klein's splendid disco-
very that the geometry of Euclid and the non-Euclidean
geometries of Lobatchewsky and Riemann are, all three, merely
different aspects of a more general kind of geometry which
includes them as special cases. The nature of these contributions
of Cayley's will be briefly indicated after we have sketched his
life and that of his friend Sylvester.
The lives of Cayley and Sylvester should be written simulta-
neously, if that were possible. Each is a perfect foil to the other,
and the life of each, in large measure, supplies what is lacking
in that of the other. Cayley's life was serene; Sylvester, as he
himself bitterly remarks, spent much of his spirit and energy
'fighting the world'. Sylvester's thought was at times as turbu-
lent as a millrace; Cayley's was always strong, steady, and
unruffled. Only rarely did Cayley permit himself the printed
expression of anything less severe than a precise mathematical
statement - the simile quoted at the beginning of this chapter
is one of the rare exceptions; Sylvester could hardly talk about
mathematics without at once becoming almost orientally
poetic, and his unquenchable enthusiasm frequently caused
him to go off half-cocked. Yet these two became close friends
and inspired one another to some of the best work that either
of them did, for example in the theories of invariants and
matrices (described later).
With two such temperaments it is not surprising that the
course of friendship did not always run smoothly. Sylvester was
frequently on the point of exploding; Cayley sat serenely on the
safety valve, confident that his excitable friend would presently
cool down, when he would calmly resume whatever they had
been discussing as if Sylvester had never blown off, while
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